Dynamics and the emergence of geometry in an information mesh
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Regular Article - Theoretical Physics
Dynamics and the emergence of geometry in an information mesh Philip Tee1,2,a 1 2
Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, AZ 85287, USA School of Engineering and Informatics, University of Sussex, Brighton BN1 9RH, UK
Received: 31 January 2020 / Accepted: 25 July 2020 © The Author(s) 2020
Abstract The idea of a graph theoretical approach to modeling the emergence of a quantized geometry and consequently spacetime, has been proposed previously, but not well studied. In most approaches the focus has been upon how to generate a spacetime that possesses properties that would be desirable at the continuum limit, and the question of how to model matter and its dynamics has not been directly addressed. Recent advances in network science have yielded new approaches to the mechanism by which spacetime can emerge as the ground state of a simple Hamiltonian, based upon a multi-dimensional Ising model with one dimensionless coupling constant. Extensions to this model have been proposed that improve the ground state geometry, but they require additional coupling constants. In this paper we conduct an extensive exploration of the graph properties of the ground states of these models, and a simplification requiring only one coupling constant. We demonstrate that the simplification is effective at producing an acceptable ground state. Moreover we propose a scheme for the inclusion of matter and dynamics as excitations above the ground state of the simplified Hamiltonian. Intriguingly, enforcing locality has the consequence of reproducing the free non-relativistic dynamics of a quantum particle.
1 Introduction 1.1 Background Reconciling General Relativity (GR) with Quantum Mechanics (QM) has not yielded a consistent and finite theory [1]. In part, this is due to the fundamentally different role that spacetime, and its geometry, plays in the two theories. In QM spacetime exists external to the theory and its geometry is input, whereas GR is essentially a theory of the geometry of spacetime. As such a quantum theory of gravity entails quana e-mails:
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tizing the geometry and a fully quantized theory of Gravity, therefore, would have to explain what it means to quantize spacetime. This is not a simple task to undertake. The original observation that quantum theory must inevitably lead to a discrete spacetime was made by Matvei Bronstein [2], but the first concrete proposal for how this could be reconciled with Lorentz invariance has its origin in the work of Hartland Snyder [3]. He proposed a framework to directly consider the implications of discretized space with a minimum length, originally as an attempt to rationalize the presence of ultra-violet cut-offs in Quantum Field Theory (QFT). The existence of a global fundamental length would seem at odds with the principle of Lorentz invariance, as observers in inertial frames moving relative to each other would disagree a
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